Maths Theorems
Important Theorems Other Theorems The topics highlighted in bold are theorems while others are properties or proofs . Trigonometric Equations # General Solutions ## sinθ = 0 ; cosθ = 0 ; tanθ = 0 Periodicity ## sinθ = sinα ; cosθ = cosα ; tanθ = tanα Factorization Formulae ## sin2θ = sin2α ; cos2θ = cos2α ; tan2θ = tan2α to cos ## acosθ + bsinθ = c , ... θ = 2nπ + α ± β [Divide throughout by √'''(a2+ b2) ] # '''Sine Rule sine of angle being changed # Cosine Rule & use Distance Formula # Projection Rule '''Cosine Rule # '''Half Angle Formulae ## sin (A/2) 1 - cosA ## cos (A/2) 1 + cosA ## tan (A/2) sin(A/2) / cos(A/2) # Area of Triangle to sine rule . Find sin C for changing C . # Hero's Formula Area of Triangle and expand sin A as 2sin(A/2)cos(A/2) & apply half angle Formulae . # Napier's Analogies sine Rule & factorization formulae Pair of Straight Lines # Combined Equation of Line through Origin multiply a1x + b1y = 0 and a2x + b2y = 0 # Converse of Theorem 1 'b=0 and b=/= 0 # '''Acute Angle between pair of straight lines 'formula of angle between two lines having slopes m1 , m2 Vectors # '''Collinearity . If ma + nb = 0 ; a & b are collinear # Coplanarity . if r = ax + by , then r,x,y are coplanar collinearity and parallelogram law # Converse of Theorem 2 # If a,b,c are 3 non coplanar vectors ''', then r can be defined as linear combination of a,b,c . # '''Section Formula ## Internal Division ## External Division # Volume of Parallelopiped # The Volume of Tetrahedron # Geometrical Applications of Vectors ## Medians of Triangle are concurrent ## Angle Bisectors of triangle are concurrent ## A Quadrilateral is a parallelogram if and only if diagonals bisect each other . ## Median of Trapezium is parallel to parallel sides and half of sum of lengths of parallel sides . ## Angle Subtended by a semicircle is a right angle ## Altitudes of triangle are concurrent 3D Geometry # l2 + m2 + n2 = 1 'as x=rcosα y=rcosβ z=rcosγ # Relation between direction cosines and direction ratios direction cosines to a constant # Acute Angle between two lines dot product formula Line # '''Equation of Line : r = a + λb 'a line AP ; draw a vector b parallel to it # 'Equation of line passing through A & B : r = (1-λ) a + λb 'lines AP and AB # Distance of a Point from line [] # Skew Lines - Shortest Distance # Distance between Parallel Lines Plane #'Equation of plane : r.n = p 'the normal vector and then dot product formula #'r.n = a.n 'product condition #Distance of a point from a Plane Differentiation # '''If f(x) is differentiable at a point , then it is also continuous at that point . is not true # Composite Functions # Inverse Functions # Derivatives of inverse trigonometric Ratios : sin-1x , cos-1x , tan-1x , cot-1x , sec-1x , cosec-1x # Parametric Functions Integration # Method of Substitution # Integration by Parts u.v 'of v = w # '''ex+ f'(x) 'with differentiation of question # Integrals : ## tan x , cot x , sec x , cosec x of Substitution ## C.T.S. divide by 2a & adjust OR x = asin , a tan ,etc. ## '√(a2-x2) , √(x2+a2) , √(x2-a2) . 'u.v Definite Integration # Reversing the limits # Substitution # Splitting of Limits # Limit 0 to a # Limit a to b # Dividing Limit 2a # Even-odd Functions '''NOTE : The topics highlighted in bold are theorems while others are properties or proofs . Category:Maths Revision